Motion of a three-phase contact line along a wet surface

The motion of the contact line in gas-liquid-solid systems is theoretically investigated for small values of the capillary number and Reynolds number. The possible existence on the solid substrate of a residual microscopic film formed by adsorbed liquid molecules is taken into account and the spreading characteristics of the liquid on dry and wet substrates are compared. It is shown that, in accordance with the experimental data, in the model employed the motion of the liquid during wetting is rolling motion, and that the increase in the dynamic contact angle is slower for a wet than for a dry substrate. The maximum dynamic contact angle is much less than 180°. The flow structure in the neighborhood of the moving contact line is analyzed and it is shown that under certain conditions regions with closed streamlines may be formed. The reason for this is the self-induced Marangoni effect — the reaction of the surface tension gradient on the liquid-solid boundary caused by the liquid flow on the flow that caused it.Based on a paper read at the Seventh Congress on Theoretical and Applied Mechanics, Moscow, August 1991. Presented by R. I. Nigmatulin.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 81–89, November–December, 1992.     

On the stability margin of equilibrium states of a capillary liquid in a slit

The stability margin is determined for symmetric equilibrium shapes of the free surface of a liquid suspended in a slit and subject to gravity and surface tension. The calculations are made in the range of variation of the parameters, the wetting angle and the Bond number, adjoining the boundary of the stability region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 88–93, July–August, 1979.     

On the shapes of swirling annular jets of a liquid

Annular jets of an incompressible liquid moving in a gas at rest are of interest for applications. A critical analysis of the investigations into jets from centrifugal nozzles is contained in [1]. These investigations elucidated the experimentally observed tulip and bubble jet shapes, and also predict the existence of annular jets of periodic shape. However, simplifications of the flow details are made to obtain the results. For example, in the equations describing the equilibrium of the forces acting on the film, no allowance is made for forces that arise on account of the curving of its shape in the meridional sections nor for the variability of the tangential velocity component in the field of the centrifugal forces. In the present paper, the method of [2] is used to derive equations that describe the flow of swirling annular jets of liquid with uniform profile of the longitudinal velocities in an undisturbed ideal medium with allowance for surface tension and gravity forces and also the pressure difference outside and within the jet. The results of calculations are given that illustrate the dependence of the jet shapes on the relative contributions of the capillary and inertial forces and also the pressure difference, the intensity of the initial swirling, the angle at which the liquid leaves the nozzle, and the gravity force.Translated from Izyestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 144–148, September–October, 1979.I am grateful to V. Ya. Shkadov for interest in the work.     

Flow dynamics of an intensively evaporating wave film of a liquid

Experimental results on the behavior of a laminar–wave film of liquid nitrogen evaporating intensively under conditions of a gravitational flow on a locally heated vertical surface are described. It was found that certain heat fluxes change significantly the shape of the residual layer and increase the relative amplitude of large waves. For the first time, data are obtained on the change in the probability density of the local film thickness as a function of the heat–flux density within the range of Reynolds numbers from 32 to 103. The effect of the heat–flux density on the phase velocity and shape of large waves is shown. Heat–flux densities at which dry spots arise were determined as functions of the streamwise coordinate of the wave film of the saturated liquid.     

Hydrodynamics in weak force fields onset of steady thermocapillary convection in a spherical fluid layer under zero-g conditions

In the study of cellular convection in an infinite plane fluid layer with a free surface, both the Archimedes and thermocapillary forces [1–3] have been cited as reasons for the onset of convection. This has also been confirmed experimentally [4], When mass forces are absent or negligibly small it is natural to pose the question of the onset of pure thermocapillary convection or convection caused only by the surface tension gradients (see [2–3]). In the present paper, this problem is examined for a spherical fluid layer under zero-g conditions.     

Determination of the equilibrium state of a capillary liquid in a vessel

This article describes a method for determination of the form of the equilibrium surface of a liquid in a given vessel of arbitrary axiosymmetric form. Capillary, gravitational, and centrifugal forces act on the liquid. Liquid volume and wetting angle are given. Curves are constructed for the case of negative overloads by a homogeneous gravitational field, which are used to find the equilibrium states. An example which illustrates the application of the method is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–7, July–August, 1973.     

Self-similar problem of the decay of a two-dimensional discontinuity

The plane problem of the decay of an arbitrary two-dimensional discontinuity for the gasdynamics equations is considered. The initial surface of the discontinuity is assumed to have the shape of an angle close to . The existence and uniqueness of the solutions of the problem in a linear formulation are proved.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 29–38, March–April, 1972.In conclusion, the author is grateful to L. V. Ovsyannikov for interest in the research and useful comments.     

Convective stability of a weakly conducting liquid in an electric field

In an inhomogeneously heated weakly conductive liquid (electrical conductivity 10^–12–1 cm^–1) located in a constant electric field a volume charge is induced because of thermal inhomogeneity of electrical conductivity and dielectric permittivity. The ponderomotive forces which develop set the liquid into intense motion [1–6]. However, under certain conditions equilibrium proves possible, and in that case the question of its stability may be considered. A theoretical analysis of liquid equilibrium stability in a planar horizontal condenser was performed in [2, 4]. Critical problem parameters were found for the case where Archimedean forces are absent [2]. Charge perturbation relaxation was considered instantaneous. It was shown that instability is of an oscillatory character. In [4] only heating from above was considered. Basic results were obtained in the limiting case of disappearingly small thermal diffusivity in the liquid (infinitely high Prandtl numbers). In the present study a more general formulation will be used to examine convective stability of equilibrium of a vertical liquid layer heated from above or below and located in an electric field. For the case of a layer with free thermally insulated boundaries, an exact solution is obtained. Values of critical Rayleigh number and neutral oscillation frequency for heating from above and below are found Neutral curves are constructed. It is demonstrated that with heating from below instability of both the oscillatory and monotonic types is possible, while with heating from above the instability has an oscillatory character. Values are found for the dimensionless field parameter at which the form of instability changes for heating from below and at which instability becomes possible for heating from above.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 16–23, September–October, 1976.In conclusion, the author thanks E. M. Zhukhovitskii for this interest in the study and valuable advice.     

Cylindrical equilibrium surfaces of a capillary liquid and their stability

We consider the plane problem of the equilibrium of a capillary surface. We study the stability of a two-dimensional surface with respect to plane and spatial disturbances. We give data which can be used for deciding the question of the stability of any symmetric equilibrium surface in a field of gravitational forces and in conditions of weightlessness. We solve the problems of the stability of a liquid in a rectangular and a sectorial channel and also the problem of the separation of a plane drop from a horizontal wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 57–68, January–February, 1976.The authors are grateful to A. D. Myshkis and A. D. Tyuptsov for their evaluation and their useful comments.     

Equilibrium of a flexible permeable shell in a supersonic gas flow

The plane steady-state regime of uniform supersonic ideal-gas flow around a uniaxial absolutely flexible open permeable shell is investigated numerically [1]. The effect of the angle of attack and the degree of nonuniformity of the permeability distribution on the aerodynamic characteristics and the equilibrium shape of the shell is determined for various methods of shell attachment. Approximate localization relations, which take into account the dependence of the distributed load on a concave permeable screen on its degree of permeability and the free-stream parameters, are formulated. An example of the application of these relations to the solution of three-dimensional problems of the equilibrium of permeable shells of the circular dome type in a supersonic flow [2] is given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 152–158, September–October, 1989.     

Three-dimensional problem of unsteady wave motions of a liquid in a region of variable depth

The three-dimensional problem of unsteady wave motions of a liquid above a plane inclined floor in the framework of a linear dispersion model was considered for the first time in [1] for the particular case =/4, where is the angle of inclination of the floor plane to the free surface of the liquid. The class of exact self-similar solutions of the problem for =/2(2m + 1), m=0, 1, 2,..., for the case of an initial perturbation of a free surface of a special type which is constant in the direction of the normal to the shoreline was found in paper [2]. The present paper is devoted to the investigation of wave motions of a liquid due to an initial perturbation of arbitrary form for the angles of inclination of the floor assumed in [2]. A complete system of eigenfunctions corresponding to the continuous and discrete spectra is found. The theorem of the expansion of an arbitrary absolutely integrable function with respect to the boundary values of the eigenfunctions is proved. An exact solution of the problem is obtained and its asymptotic analysis is carried out.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 104–112, March–April, 1986.     

Standing waves of finite amplitude on surface of ideal liquid of finite depth

The nonlinear problem of plane gravitational standing waves of finite amplitude on the surface of an ideal incompressible liquid of infinite depth has been solved analytically [1], It was also solved in [2] using a new method, in which the dimensionless velocity potential, the profile of the free surface, and the frequency were expressed as power series in the parameter , equal to the ratio between the the amplitude and the wavelength. The results of these two papers agree. Below, the method of [2] is used to study plane standing surface waves of finite amplitude on the surface of a liquid of finite depth. The frequency, the profile of the free surface, and the velocity potential are expressed as power series in the small parameter . The solution is obtained in the third approximation. An expression for the amplitude dependence of the frequency is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 38–43, March–April, 1978.     

Boundary layer transition on the surface of a delta wing in supersonic flow

Laminar-turbulent transition on the surface of a delta wing has been experimentally investigated in a supersonic wind tunnel at Mach numbers M_t8=3–5. It is shown that when M,=3, Re_L=6.5·10⁶, and =–5.5° much of the upper surface of the wing in the neighborhood of the line of symmetry is occupied by a wedge-shaped region of turbulent flow. In this region the heat fluxes reach the same values as at the heat transfer maxima induced here by separated flows and may exceed the turbulent heat flux level on the windward surface of the wing. Changing the shape of the under surface of the wing from plane to pyramidal leads to acceleration of the boundary layer transition on the under surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 87–92, May–June, 1989.     

Numerical study of convection of a liquid heated from below

The equilibrium of a liquid heated from below is stable only for small values of the vertical temperature gradient. With increase of the temperature gradient a critical equilibrium situation occurs, as a result of which convection develops. If the liquid fills a closed cavity, then there is a discrete sequence of critical temperature gradients (Rayleigh numbers) for which the equilibrium loses stability with respect to small characteristic disturbances. This sequence of critical gradients and motions may be found from the solution of the linear problem of equilibrium stability relative to small disturbances. If the temperature gradient exceeds the lower critical value, then (for steady-state heating conditions) there is established in the liquid a steady convective motion of a definite amplitude which depends on the magnitude of the temperature gradient. Naturally, the amplitude of the steady convective motion cannot be determined from linear stability theory; to find this amplitude we must solve the problem of convection with heating from below in the nonlinear formulation. A nonlinear study of the steady motion of a liquid in a closed cavity with heating from below was made in [1]. In that study it was shown that for Rayleigh numbers R which are less than the lower critical value R_c steady-state motions of the liquid are not possible. With R>R_c a steady convection arises, whose amplitude near the threshold is small and proportional to (R–R_c)^1/2 (the so-called soft instability)-this is in complete agreement with the results of the phenom-enological theory of Landau [2, 3].Primarily, various versions of the method of expansion in powers of the amplitude [4–8] have been used, and, consequently, the results obtained in those studies are valid only for values of R which are close to R_c, i. e., near the convection threshold.It is apparent that the study of developed convective motion far from the threshold can be carried out only numerically, with the use of digital computers. In [9, 10] the numerical methods have been successfully used for the study of developed convection in an infinite plane horizontal liquid layer.The present paper undertakes the numerical study of plane convective motions of a liquid in a closed cavity of square section. The complete nonlinear system of convection equations is solved by the method of finite differences on a digital computer for various values of the Rayleigh number, the maximal value exceeding by a factor of 40 the minimal critical value R_c. The numerical solution permits following the development of the steady motion which arises with R>R_c in the course of increase of the Rayleigh number and permits study of the oscillatory motions which occur at some value of the parameter R. The heat transfer through the cavity is studied. The corresponding linear problem on equilibrium stability is solved approximately by the Galerkin method.     

Asymptotic form of the free surface of a uniformly rotating liquid for large bond numbers

In [1–5] boundary-layer methods were used to solve problems concerned with the equilibrium and motion of a liquid with surface tension in a strong gravitational field (for large Bond numbers Bo). In the present paper we apply these methods to problems involving the equilibrium shape of a uniformly rotating liquid, contained in a cylindrical container of arbitrary cross section or in a container which is a surface of revolution about the z axis. Both of these problems reduce to the asymptotic integration of an equation with a small parameter involving a quasilinear elliptic operator with a nonlinear boundary condition. In the second case, owing to radial symmetry, the equation for the problem goes over into an ordinary equation; however, the wetted boundary is not known beforehand. This boundary, together with the equilibrium shape, is also determined asymptotically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–12, November–December, 1973.The authors thank L. A. Slobozhanin for his help in the preparation of this paper.     

Interaction of a magnetic liquid with a conductor containing current and a permanent magnet

The present study investigates the force exerted by a magnetic liquid on a conductor with a current or a permanent magnet located near its infinite free surface. It is shown that this force is equal to the weight of the liquid raised by the magnetic field above the original horizontal level. The force is found for the cases when the liquid is weakly magnetic and when the shape of its surface differs little from plane. Consideration is given to the equilibrium of a magnet suspended on a small spring near the surface of the magnetic liquid. The critical height is found at which the magnet ceases to be held by the spring and is torn off into the liquid. The experimentally obtained values of the magnitude of of the force acting on the magnet and the height of collapse of the magnet into the liquid are in good agreement with the theoretical results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 176–181, March–April, 1988.The authors are grateful to V. V. Gogosov for valuable discussion of the study.     

One case of the explosion of a symmetric variable-thickness surface charge

Within the framework of the model of explosion phenomena proposed by M. A. Lavrent'ev, the plane problem of determining the shape of the excavation during the explosion of a constant-thickness surface charge is solved (when the pulse pressure is constant over the width of the charge). The problem of determining the shape of the excavation during the explosion of a surface charge whose section thickness varies linearly is considered below for the same model of explosion phenomena called solid-liquid [2]. (The problem reduces to a homogeneous linear Hilbert boundary-value problem with discontinuous coefficients. The solution is obtained in closed form, and recommendations are given for its practical realization.)Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 154–161, May–June, 1976.     

Dynamical equations for a liquid jet

Equations are described for the three-dimensional motion of a thin jet of viscous liquid. The jet is a liquid body whose transverse dimensions are small compared with the other characteristic dimensions of the problem. The aim in the present paper is to establish a closed system of asymptotic equations for the dynamics of such a thin jet. A more detailed derivation of quasi-one-dimensional asymptotic equations for the dynamics of thin liquid jets and an analysis on the basis of them of the curved decay shape of the jet in the linear and nonlinear stages is contained in [1, 2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 11–18, September–October, 1980.     

Axisymmetric spreading of a heavy liquid over a plane

A study is made of the steady flow over a horizontal plane of a heavy inviscid incompressible liquid which flows through the side surface of a circular cylinder which rises above the plane to height h and has a base radius ofa. The motion of the liquid is assumed to be symmetric with respect to the axis of the cylinder; the pressure p is constant (equal to the atmospheric pressure) on the free surface of the liquid. Fora/h = 1, this problem can be regarded as a problem of perturbation of the flow from a flat source by a free surface. Investigation showed that this perturbation problem is essentially nonlinear, and a solution of it in the complete region occupied by the liquid can be obtained only in variables of the boundary layer type. The problem admits linearization under the additional assumption that the parameter = Q²/(8²ga³) is small; here, Q is the constant volume flow rate of the liquid per unit height of the cylinder, and g is the acceleration of free fall. For the case 1, 1 the problem is solved by the method of integral transformations. A noteworthy feature of the solution is the slow damping of the perturbations of the velocity with the depth (inversely proportional to the square of the distance from the free surface), in contrast to the similar problem of the wave motions of a heavy liquid, for which the velocity perturbations are damped exponentially.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–7, March–April, 1984.     

Dependence of the stability criterion of nucleate boiling on the compressibility of the vapor

The article gives the results of a study of the motion of bubbles and their deformation near the heating surface at different pressures. It was observed that, during the time of their growth, the gaseous medium in the bubbles is in a compressed state.Nomenclature R) radius of bubble - R_h) maximul radius of a deformed bubble in the horizontal plane - R_v) maximal radius of a deformed bubble in the vertical plane - ) specific weight - B) universal gas constant - ) surface-tension coefficient - p) pressure - ) edge wetting angle - g) acceleration due to gravity - V) volume - ) molecular weight - C_T) isothermal velocity of sound Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 77–81, July–August, 1971.